An arithmetic count of the lines on a smooth cubic surface

Journal Article (Journal Article)

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field, generalizing the counts that over there are lines, and over the number of hyperbolic lines minus the number of elliptic lines is. In general, the lines are defined over a field extension and have an associated arithmetic type in. There is an equality in the Grothendieck-Witt group of, where denotes the trace. Taking the rank and signature recovers the results over and. To do this, we develop an elementary theory of the Euler number in-homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.

Full Text

Duke Authors

Cited Authors

  • Leo Kass, J; Wickelgren, K

Published Date

  • January 1, 2021

Published In

Start / End Page

  • 677 - 709

International Standard Serial Number (ISSN)

  • 0010-437X

Digital Object Identifier (DOI)

  • 10.1112/S0010437X20007691

Citation Source

  • Scopus